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Edited by: Hannah Wexler, University of California, Los Angeles, United States

Reviewed by: Roger Garrett, University of Copenhagen, Denmark; Edze Westra, University of Exeter, United Kingdom

This article was submitted to Microbial Physiology and Metabolism, a section of the journal Frontiers in Microbiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

All cellular organisms coevolve with multiple viruses, so that both virus-host and intervirus conflicts are major factors of evolution. Accordingly, hosts evolve multiple, elaborate defense systems and viruses respond by evolving means of antidefense. Although less thoroughly characterized, several dedicated mechanisms of intervirus competition have been described as well. Recently, the genomes of some bacterial and archaeal viruses have been shown to harbor CRISPR mini-arrays that typically contain a single spacer targeting a closely related virus. The involvement of mini-arrays in an intervirus conflict has been experimentally demonstrated for a pair of archaeal viruses. We model the evolution of virus-encoded CRISPR mini-arrays using a game theoretical approach. Analysis of the model reveals multiple equilibria that include mutual targeting, unidirectional targeting, no targeting, cyclic polymorphism, and bistability. The choice between these evolutionary regimes depends on the model parameters including the coinfection frequency, differential productivity of the conflicting viruses, and the fitness cost of mini-arrays. At high coinfection frequencies, the model becomes a version of the Prisoner’s dilemma in which defection, i.e., mutual targeting between the competing viruses, is the winning strategy.

Most microbial communities are associated with highly diverse and abundant viral populations (

Analysis of viral genomes from databases and environmental samples has revealed that viruses can co-opt parts of the CRISPR-Cas antiviral defense systems from the host and use them as weapons against competing viruses. Specifically, some viruses and prophages contain mini-CRISPR arrays with 1 or 2 spacers that target sequences from related viruses, typically infecting the same host and sometimes isolated from the same environment (

We were interested in investigating the role of mini-arrays in interviral conflicts from an evolutionary cost vs. benefit perspective. To this end, here we develop a game-theoretical model of interviral competition mediated by mini-CRISPR arrays. The main goal of this work is to identify the conditions under which such virus-against-virus targeting becomes a successful strategy. Our results shed light on the ecological and evolutionary trade-offs that lead to the engagement of bacterial and archaeal viruses in CRISPR-mediated arms races. We provide testable predictions on the differential prevalence of mini-CRISPR arrays among viruses with different lifestyles.

We present a minimal model of interviral conflict mediated by mini-CRISPR arrays. Because interviral conflicts are driven by mixed coinfection of hosts, the coinfection probability is the main ingredient of our model. Below, we introduce a parameterization of the coinfection probability based on Bayes’ theorem and discuss it in detail. Then, we describe the mathematical model, which combines elements from game theory and ordinary differential equations, and obtain its steady state solutions for a general case. We apply the model to three scenarios that involve, respectively, two lytic viruses; a lytic virus and a temperate virus (or a provirus); and two temperate viruses or proviruses. Finally, we briefly discuss some extensions of the model to account for the loss of CRISPR spacers and the competition among more than two viruses.

Let us consider two viruses,_{B—A} is the probability that virus _{A—B}, as the probability that virus _{B—A} can be understood as the conditional probability of finding a host infected by virus

where _{A} and _{B} are the probabilities of finding hosts infected by virus _{B—A} and _{A—B} can take different values. Such differences are caused by differences in abundance, infectivity, spatial distribution, host range, and any other factor that introduces an imbalance in the prevalence of viral infections. All other factors being equal, the coinfection probability will be greater for the virus with the lower abundance, the more restricted distribution, or the narrower host range.

In the case of viruses that target each other via mini-CRISPR arrays, the outcome of a coinfection event likely depends on which virus infected the host first. To account for that, we split each coinfection probability into two components, that is, _{B|A} = _{AB|A} + _{BA|A} and _{A|B} = _{AB|B} + _{BA|B}, where the subindices _{A} and _{B} to denote the fractions of the coinfection events in which viruses _{AB|A} = _{A}×_{B|A} and _{BA|B} = _{B}×_{A|B}.

We model the competition among two groups of viruses (species or strains), each of which can possess or lack mini-CRISPR arrays against the other. For simplicity, and without loss of generality, we normalize the productivity of an infection by a single virus that lacks mini-CRISPR arrays to make it equal to 1 (this is an arbitrary choice because multiplying all productivities by the same factor does not affect the relative composition of the population). When two viruses coinfect the same host, the offspring of each virus depends on whether they possess or lack mini-CRISPR arrays targeting each other:

In the absence of such arrays, the average productivity of each virus in a mixed coinfection is given by the parameter _{A} for virus _{B} for virus _{A} = _{B} = 1/2. By using different values of _{A} and _{B}, the model can account for unequal competitive abilities (_{A}≠_{B}), interference (_{A} + _{B} < 1) and facilitation (_{A} + _{B} > 1) between viruses.

If only one virus has a mini-CRISPR array with a spacer against the other virus, the productivities are _{A} for virus _{B} for virus

If both viruses target each other, we assume that “the first that arrives wins.” This is biologically plausible because, whereas transcription and processing of mini-CRISPR arrays require some time, scanning and cleavage of the targeted viral DNA is very fast (_{A}, _{A}) and _{B}, _{B}). As explained above, _{A} can be interpreted as the probability that virus _{B}). Note, however, that the “first who arrives wins” assumption can be easily relaxed to allow for some degree of mutual destruction upon coinfection, in which case _{A} + _{B} < 1.

The parameters of the model are summarized in

Variables and parameters of the model.

_{B—A} |
Coinfection probability for virus |

_{A—B} |
Coinfection probability for virus |

_{A},_{B} |
Cost of the mini-CRISPR array |

_{A},_{B} |
Viral productivity in a mixed coinfection |

_{A},_{B} |
Probability that the virus survives when there is mutual targeting |

_{A},_{B} |
Fraction of the viral population that harbors a mini-CRISPR array |

^{∗},^{∗∗} |
Critical coinfection probabilities for the maintenance of a mini-CRISPR array |

Mean productivity of a virus (_{B—A}, depending on the presence or absence of mutually targeting CRISPR arrays.

– | |||

Anti- |
+ | (1-_{A})(1-_{B|A} + _{A}_{B|A}) |
1-_{A} |

– | 1-_{B|A} |
1−(1−_{A})_{B|A} |

Before proceeding with the formal analysis of the system, some immediate insight can be gained from inspecting the fitness matrix of the evolutionary game. If we consider a scenario in which virus _{B|A} > _{A}/(1−_{A}). Conversely, if the entire population of virus _{B|A} > _{A}/(_{A}−_{A}_{A} + _{A}). Thus, the outcome of the evolutionary process is determined by a trade-off between the cost of the CRISPR mini-array and the potential benefit conferred by the spacer against the competing virus during coinfections.

To analyze the general case of mixed populations, let us introduce new variables _{A} and _{B}, both with values between _{A} and 1−_{B} are the relative sizes of the spacer-free subpopulations. Focusing on strain _{A,1} and _{A,2}, respectively) results from multiplying the fitness matrix in

Because of the complexity of virus-host interaction networks in natural microbial communities, it is unlikely that the host ranges or geographical distributions of two viruses completely overlap (

where ϕ_{A} is the mean fitness of strain

The mean fitness (ϕ_{B}) and class–specific fitness values (_{B,1} and _{B,2}) for strain

where

A comprehensive analysis of the solutions of Eq. 4 indicates that the outcome of the evolutionary arms race between the two viruses is given by the relative values of the coinfection probabilities _{B—A} and _{A—B} with respect to two pairs of critical thresholds,

(note that we omit the subindices ^{∗}) or when it does (^{∗∗}). Depending on the coinfection rate, the evolutionary arms race can lead to 5 qualitatively different regimes (

Outcomes of the mini-CRISPR array-mediated virus arms race as a function of the coinfection probabilities experienced by each virus. Depending on the relative ordering of the critical thresholds ^{∗} and ^{∗∗}, three qualitatively different cases exist _{A} = 0.8, _{B} = 0.5, _{A} = 0.5, _{B} = 0.2, _{B|A} = 0.2, _{A|B} = 0.3.

No targeting if

Unidirectional targeting, with

Unidirectional targeting, with

Mutual targeting if

Cyclic dominance, with an alternation of targeting and non-targeting subpopulations, if _{B—A} <_{A—B} <_{B—A} <_{A—B} <

Two important observations are pertinent. First, with the exception of the last regime, the population always reaches an equilibrium state in which all viruses of a given type (_{B—A} <_{A—B} <_{B—A} <_{A—B} <

It follows from Eq. 9 that, in order to maintain a mini-CRISPR array, the probability of coinfection must be greater than the cost of the array. Moreover, possession of mini-CRISPR arrays is more advantageous for strains with low yields in mixed coinfections (low _{A}) because those would benefit the most from curtailing other, more efficient coinfecting strains.

When considering 2 lytic viruses, the model can be simplified by noting that any of them can be the first to enter the host cell with the same probability, that is, _{A} = _{B} = _{A} = _{B} =

The dependencies of these thresholds on the fitness cost and the productivity in mixed coinfections are plotted in

Dependency of the critical coinfection thresholds ^{∗} (red) and ^{∗∗} (blue) on the cost of the mini-CRISPR array

If both viruses can use the molecular machinery synthesized by the other virus, and in the absence of other sources of interference, the productivity of each virus in a mixed coinfection will be half of the productivity in a pure infection (that is, ^{∗∗}≈^{∗} =

Outcomes of the mini-CRISPR array-mediated virus arms race in the case of two lytic viruses, whose productivity in mixed coinfections is half of their productivity in a pure, single infection. Each virus will maintain mini-CRISPR arrays targeting the other virus if the probability of coinfection is greater than twice the cost of the array. Viruses

Let us now consider a scenario with one lytic (_{L} = 0 and _{T} = 1. Note, however, that unidirectional targeting of the temperate virus by the lytic virus leads to the elimination of the temperate virus and production of pure offspring of the lytic virus. If the temperate virus is present as a provirus, its destruction will also lead to the degradation of the host genome; in that case, we assume that replication of the lytic virus remains unaffected by host genome degradation. Because the cost of carrying a spacer can differ among viruses with distinct lifestyles, we consider virus-specific cost parameters, _{L} and _{T}. Provirus induction is implicitly modeled by the parameter _{T}, which jointly represents the probability of induction and the fraction of viral particles that successfully insert as a provirus in a new host.

As in the previous cases, the outcome of the evolutionary process depends on the relative values of the coinfection probabilities _{L—T} and _{T—L} with respect to three critical thresholds:

(there is an additional threshold for the lytic virus at _{T|L}≤1). Once again, ^{∗} and ^{∗∗} represent the critical coinfection probabilities beyond which it becomes profitable to target the other virus that either does (^{∗∗}) or does not engage (^{∗}) in the arms race via its mini-CRISPR array.

Evolutionary outcomes for the arms race between a lytic virus (_{T—L} and _{L—T}. Black borders denote the presence of a mini-CRISPR array. The expressions for the critical coinfection thresholds

The outcome of the competition between a lytic and a temperate virus depends on (i) whether the latter is an actively replicating non-lytic virus or a provirus. For a provirus, we must distinguish (ii) whether or not it provides the host with CRISPR-independent immunity against subsequent infection by the lytic virus, and (iii) whether such immunity can be overcome by a lytic virus that carries a mini-array against the temperate virus. Let us start with the case of an actively replicating non-lytic virus. Because the long-term productivity of chronic viruses is tightly linked to the survival of the host, superinfection by lytic viruses severely reduces their fitness (_{T}≪1). In terms of the model, this implies that temperate viruses can maintain mini-arrays against lytic viruses as long as the coinfection probability is greater than the cost of the mini-array (_{T}). Thus, frequent encounters between the temperate and the lytic virus will result in unidirectional targeting from the former toward the latter. The same reasoning applies to those proviruses that do not provide immunity against a given lytic virus. However, many proviruses encode superinfection exclusion mechanisms that prevent infection by closely related viruses and, sometimes, even by distantly related ones (_{L} = 0, _{T}≈1), and leads to a cyclic targeting dynamics (no targeting, lytic to provirus, mutual targeting, provirus to lytic, and no targeting again) if the coinfection probabilities for both viruses are greater than the cost of the mini-array (_{T}, _{L}).

Let us consider a scenario in which a temperate virus (

The model for the competition between two actively replicating temperate viruses is formally the same as for two lytic viruses, with the critical coinfection thresholds given by Eqs 11–12 (_{A})(1-_{B|A} + _{A}_{B|A}), regardless of whether virus ^{∗} for the competition of 2 proviruses becomes

whereas ^{∗∗} remains the same as in Eq. 12.

For both proviruses and actively replicating non-lytic viruses, the possible evolutionary outcomes will depend on the effect of coinfection for the long-term productivity of the temperate virus. In the case of actively replicating non-lytic viruses, it seems reasonable to assume that coinfection substantially decreases viral productivity. To get an approximate idea of how that affects the evolutionary outcome, if the productivity of the resident virus decreases to half of its original value, the evolutionary regimes will coincide with those shown for lytic viruses in ^{∗∗}≈2^{∗}≥1. Under this assumption, we would expect either no targeting or mutual targeting between proviruses (but not unidirectional targeting), with mutual targeting evolving only if the coinfection probability of both viruses is greater than twice the cost of the mini-array. The same outcomes are predicted if one or both proviruses possess superinfection exclusion mechanisms that can be overcome through CRISPR-mediated targeting (this scenario corresponds to setting

In the model discussed so far, the only factor that opposes the engagement of viruses into CRISPR-mediated arms races is the fitness cost of mini-arrays. Motivated by the fact that prokaryotic genome evolution displays a consistent deletion bias (

Mini-array loss can be easily incorporated into the present framework as follows. If spacers are lost at a fixed rate

with the same fitness parameters _{A,1}, _{A,2}, _{B,1}, _{B,2}, ϕ_{A}, ϕ_{B} as before. For simplicity, we present the equations for

The general expression for the second critical threshold becomes more complicated because it indirectly depends, for a given virus, on the coinfection probability for the other virus. Specifically,

where

In these expressions, _{B} is the relative reduction in the fitness of virus _{A} by replacing subindices _{A—B} and _{B—A}, the critical coinfection thresholds _{A—B} vs. _{B—A} parameter space. Nevertheless, the parameter combinations that lead to no targeting, unidirectional targeting, and mutual targeting can still be obtained by applying the rules from _{A}, _{B}),

The expressions from Eqs 18–20 fully apply to a loss-driven scenario where the mini-CRISPR array entails no cost (_{A|B}, 2_{B|A} and it is expected to hold if coinfections are frequent enough to promote the maintenance of the mini-CRISPR array). These considerations lead to the critical thresholds ^{∗} = ^{∗∗} =

To summarize, the qualitative results of the simple model are robust to the introduction of modest rates of spacer loss. An obvious quantitative difference is that, because loss of spacers continuously produces spacer-free viruses, the populations that engage in the arms race will be polymorphic. For example, in a unidirectional targeting scenario where strain _{A} = 1−_{A} of virus

Competition among bacterial and archaeal viruses can lead to a mini-CRISPR array-mediated interviral arms race if coinfections are frequent enough. The model described here predicts how frequent coinfections must be so that it becomes advantageous for a virus to carry a mini-CRISPR array targeting a competitor. Precise quantification of such threshold requires measuring the values of two key parameters: (i) the cost of maintaining a mini-CRISPR array with the virus-targeting spacer, and (ii) the productivity of a virus lacking a mini-CRISPR array in a mixed coinfection relative to the productivity of that same virus in isolation. For two lytic viruses, a necessary (and often sufficient) condition for the emergence of mutual targeting is that the coinfection probability is greater than twice the cost of having the array. The same condition holds for the emergence of mutual targeting between two temperate viruses (either proviruses or actively replicating non-lytic viruses). For the competition between lytic and temperate viruses, the model predicts evolution of unidirectional targeting, from the temperate virus to the lytic virus, or cyclic targeting dynamics if the coinfection probability exceeds the cost of the mini-array. Mutual targeting between lytic and temperate viruses is unstable because lytic viruses only benefit from targeting a temperate virus if the latter does not already target the lytic virus. These results emphasize the interplay between superinfection exclusion and CRISPR-mediated targeting in interviral conflicts that involve prophages. Indeed, the outcome of the mini-CRISPR array-mediated arms race critically depends on whether the competing viruses encode mechanisms to prevent superinfection and whether such mechanisms can be vanquished by viruses that carry mini-CRISPR arrays.

Our analysis sheds light on the effect of viral interference and facilitation on interviral conflicts, and predicts that mini-CRISPR arrays are more likely to evolve in pairs of closely related viruses that strongly interfere with each other when coinfecting the same host cell. Another prediction that arises from the model is that mini-CRISPR arrays should be more frequent in “specialist” viruses (viruses with narrow host and geographical ranges) than in “generalist” ones, especially if the virus-host interaction network is nested as suggested by empirical data (

Empirically observed instances of virus pairs with mutually targeting CRISPR-arrays often involve closely related viruses (based on the similarity of the terminase large subunit) with high sequence identity in their protospacer regions (

The model described here is simple, yet capable to produce a broad variety of evolutionary outcomes. Specifically, equilibrium states with mutual targeting, unidirectional targeting, no targeting, cyclic polymorphism, and bistability are recovered. The asymmetry that leads to unidirectional targeting most often results from different coinfection probabilities for each virus. Thus, if virus

The model can be readily generalized to include three or more competing viruses with different coinfection probabilities. In that case, the outcome of the competition depends on how the fitness cost scales with the number of targeted viruses. In particular, if the cost increases linearly, the problem can be reduced to a combination of pairwise interactions, each of which can be separately analyzed with the 2-virus model. More specifically, the equations of the 2-virus model can be simply applied to each pair of viruses to determine if they carry spacers against the other. A notable aspect of the intervirus arms race becomes manifest in the regime of frequent coinfections with mutual targeting. In this regime, viruses engage in an analog of the Prisoner’s Dilemma (

Notwithstanding the possible sampling bias, most empirical instances of viral mini-CRISPR arrays targeting other viruses involve prophages or non-lytic archaeal viruses. Because the formal conditions required for the evolution of mini-CRISPR arrays in lytic and temperate viruses are not dramatically different, we propose that the higher prevalence of mini-CRISPR arrays in temperate viruses results from a greater frequency of coinfection in these viruses. Indeed, single-cell sequencing of a hyperthermophilic archaeal community has recently shown that most archaeal cells harbor one or more viruses (

From an evolutionary perspective, our model addresses the role of selection and neutral loss in the fate of mini-CRISPR arrays. However, it does not include other sources of genotypic variation that fuel the evolutionary process, such as mini-array acquisition and mutation of target sequences. The acquisition of spacers was not included in our model because of the uncertainty about the mechanism and also because it is unclear whether spacer capture is a limiting factor for the observed distribution of mini-CRISPR arrays in viral taxa. This aspect will need to be revisited as more empirical evidence becomes available. Another relevant process that is not covered in this work is the evolutionary escape from CRISPR targeting through mutations in viral proto-spacers. Because the long-term efficacy of proto-spacer mutation as an escape strategy depends on the rate at which new spacers are acquired, the design of more realistic coevolutionary models will also require a better understanding of the dynamics of spacer and mini-array acquisition in viral populations. Further refinements to the model should also include an explicit consideration of the host and virus population dynamics and how changes in abundance reflect on coinfection probabilities. Such an explicit modeling of time-dependent coinfection dynamics, which is not part of the present model, can shed light on whether the larger fluctuations of host populations associated with the proliferation of lytic viruses lead to fundamental differences in the fate and efficiency of mini-CRISPR arrays, as predicted for “regular,” cell-borne arrays (

There could be additional potential benefits that mini-CRISPR arrays might provide to the carrier virus and that are independent of interviral conflict but rather have to do with overcoming the host defenses. For example, high levels of transcription of the mini-CRISPR array could potentially lead to competitive exclusion of the host spacers from the CRISPR-Cas effector modules, protecting the virus from targeting. Additionally, repeats in the mini-CRISPR array could promote the insertion of the virus within the host CRISPR array, thus impairing the host immune response; indeed, insertion of proviruses into CRISPR arrays has been observed in several bacterial genomes (

Finally, it is worth noting that the modeling framework developed here can be generalized beyond viruses, to plasmids that might carry mini-arrays (

All datasets for this study are included in the article/supplementary material.

JI and GF conceived the study. JI constructed the model. JI, GF, YW, and EK analyzed the results. JI and EK wrote the manuscript that was edited and approved by all authors.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.